A364230 Triangle read by rows: T(n, k) is the number of n X n symmetric Toeplitz matrices of rank k using all the integers 1, 2, ..., n.

1, 0, 2, 0, 0, 6, 0, 0, 0, 24, 0, 0, 0, 0, 120, 0, 0, 0, 0, 2, 718, 0, 0, 0, 0, 4, 31, 5005, 0, 0, 0, 0, 0, 2, 44, 40274, 0, 0, 0, 0, 0, 0, 4, 272, 362604, 0, 0, 0, 0, 0, 0, 0, 111, 774, 3627915, 0, 0, 0, 0, 0, 0, 2, 14, 244, 6974, 39909566, 0, 0, 0, 0, 0, 0, 0, 4, 64, 743, 9533, 478991256

Offset

1,3

Links

Table of n, a(n) for n=1..78.

Wikipedia, Toeplitz Matrix

Example

The triangle begins:
  1;
  0, 2;
  0, 0, 6;
  0, 0, 0, 24;
  0, 0, 0,  0, 120;
  0, 0, 0,  0,   2, 718;
  0, 0, 0,  0,   4,  31, 5005;
  ...

Mathematica

T[n_, k_]:= Count[Table[MatrixRank[ToeplitzMatrix[Part[Permutations[Range[n]], i]]], {i, n!}], k]; Table[T[n, k], {n, 8}, {k, n}]//Flatten

Prog

(PARI)
MkMat(v)={matrix(#v, #v, i, j, v[1+abs(i-j)])}
row(n)={my(f=vector(n)); forperm(vector(n, i, i), v, f[matrank(MkMat(v))]++); f} \\ Andrew Howroyd, Dec 30 2023

Crossrefs

Cf. A000142 (row sums), A350953 (minimal determinant), A350954 (maximal determinant), A351019 (minimal permanent), A351020 (maximal permanent), A356865 (minimal nonzero absolute value determinant), A364231 (right diagonal).

Sequence in context: A106512 A181229 A364233 * A259857 A364790 A094785

Adjacent sequences: A364227 A364228 A364229 * A364231 A364232 A364233

Keyword

nonn,tabl

Author

Stefano Spezia, Jul 14 2023

Extensions

Terms a(46) and beyond from Andrew Howroyd, Dec 30 2023

Status

approved